Dynamical Analysis and Periodic Solution of a Chaotic System with Coexisting Attractors

Chaotic attractors with no equilibria, with an unstable node, and with stable node-focus are presented in this paper. The conservative solutions are investigated by the semianalytical and seminumerical method. Furthermore, multiple coexisting attractors are investigated, and circuit is carried out. To study the system’s global structure, dynamics at inﬁnity for this new chaotic system are studied using Poincar´e compactiﬁcation of polynomial vector ﬁelds in R 3 . Meanwhile, the dynamics near the inﬁnity of the singularities are obtained by reducing the system’s dimensions on a Poincar´e ball. The averaging theory analyzes the periodic solution’s stability or instability that bifurcates from Hopf-zero bifurcation.


Introduction
Although many chaotic systems were constructed, many of them have focused on the type of equilibria, simple algebraic structure, the number of equilibria, or specific features in attractors. However, the systems' global dynamics were rarely analyzed. Moreover, the dynamics mainly rely on numerical methods; qualitative analysis is rare. erefore, it is incredibly interesting to qualitatively analyze the dynamics such as infinity dynamics and periodic solutions. In [43], Sprott proposed three standards (Feasibility of Study, Simple system, and Complex dynamic behavior) for new chaotic systems to be published and used these criteria, and motivated by undiscovered features of systems with coexisting attractors, a following new chaotic system is proposed in this paper: In comparison with the other well-known chaotic systems, the unique characteristics of the proposed system are as follows. (a) e chaotic system can have an unstable node, a stable node-focus, and no equilibria. (b) e proposed system's chaotic attractor is conservative for some particular parameters but dissipative for others. e conservative nature of the new systems is proved by the semianalytical and seminumerical method. (c) ree coexisting attractors appear in the new system. (d) Infinity dynamics and periodic solutions are analyzed based on Poincaré compactification and the averaging method. Compared with the numerical method, these qualitative analysis methods help understand this new system's global structure.
It can be seen from these characteristics that the new system has rich dynamical properties. erefore, it is necessary to explore the dynamical properties to analyze the global dynamics. e rest of the paper is organized as follows. In Section 2, chaotic attractors with no equilibria, unstable node, and stable node-focus are presented. e conservative solutions are investigated by four methods: (a) the sum of finite-time local Lyapunov exponents is zero; (b) divergence of the vector field is zero; (c) local Lyapunov dimension is equal to the order of the system; and (d) the Hamiltonian energy of the system is invariable. Furthermore, multiple coexisting attractors and analog circuit simulation are also shown in this section. In Section 3, to research the system's global structure, by reducing the system's dimensions on a Poincaré ball via the theory of Poincaré compactification, the infinity dynamics of the system are obtained. Also, an analytic proof of the existence of the Hopf-zero bifurcation is provided. e periodic solution's stability or instability, born in Hopf-zero bifurcation, is analyzed by the averaging theory in Section 4. Finally, concluding remarks are given in Section 5.

Dynamical Analysis
Let _ x � 0, _ y � 0, and _ z � 0; system (1) has a line of equilibria at (0, 0, z) when a � 0 and e � 0 or has one equilibrium E(0, 0, (− a/e)) when e ≠ 0 or has no equilibrium when a ≠ 0 and e � 0. When e � 0, the conservativeness of system (1) can be testified by the divergence as follows: e conservativeness is not obvious since the dissipation is given by the time averaged value of − z(t) along the trajectory. (1) with No Equilibrium. System (1) is rotationally symmetric under the transformation (x, y, z) ⟶ (− x, − y, z). So, any attractors are either rotationally symmetric around the z-axis or there is a symmetric pair of them. Figure 1 shows the attractors of system (1) with the initial data as (0, 5, 0), the parameters b, c, and e are fixed, and the parameter a is changed (a � − 6.17, a � − 4.5, and a � − 0.8), using MATLAB R2020b simulation software, and its corresponding attractors with their properties are summarized in Table 1. en, calculate the Lyapunov exponents spectrum of system (1) for fixed parameter b, c, and e, and let a varies. When a ∈ (− 7, 0), the local finite-time Lyapunov exponents spectrum and the largest Lyapunov exponent of system (1) are shown in Figure 2.

Chaos in the Studied System
According to Figure 2, when a ∈ (− 7, 0) and b � − 1, c � 1, and e � 0, the largest Lyapunov exponent trend can be observed. is observation is also verified by Table 1.
Following the definition of average value of the variable s(t) by s(t) � lim t⟶∞ ( t t 0 s(t)dt/t − t 0 ) as stated in Ref. [44], the average of z(t) of system (1) with special parameters is shown in Table 1, system (1) has a conservative solution since the sum of finite-time local Lyapunov exponents is zero, and the average of z(t) of system (1) with special parameters is shown in Figure 3. e plots of average of z(t) are found to be equal to zero as in Figure 3. e results are not shown here to avoid repetition of the figures. e average values of z(t) of the system are zero. In fact, when b � 0, c � 1, and e � 0, system (1) is a special case of the Nose-Hoover oscillator [45,46]. It is invariant to the transformation t ⟶ − t; hence, it is time-reversible with LEs that are symmetric around zero. Furthermore, we can provide a positive definite energy function in a quadratic form as the Hamiltonian function: Taking the time derivative of H, we have which shows that the energy of system (1) is invariable if z(t) is zero. erefore, the system has a conservative nature (in Figure 4(a), see a conservative torus). It also implies that system (1) has coexisting dynamics, which are analyzed in the following section.

Coexisting Attractors.
In Section 2.1, we have seen that system (1) shows chaotic dynamics with no equilibria in e � 0. Moreover, since the sum of Lyapunov exponents is 2 Complexity zero for system (1) in special parameters, the conservative solution and chaotic flows are coexisting in this system (in Figure 4(a), chaotic attractor coexists with a torus). Indeed, we can see that there exits a limit cycle for some initial points a − bx 2 + cy 2 � 0 and z � 0 in a < 0, − b � c � 1, and e � 0. Following these conditions, we can see that if an initial point starts form a limit cycle x 2 + y 2 � a and z � 0, we have _ z � 0, and the state variable z is fixed. Since z ≡ 0, hence, system (1) will become System (5) is a Hamiltonian system (conservative). All orbits start from x 2 + y 2 � a and stay in this circle. So, we claim that system (1) has coexisting attractors under condition a < 0, − b � c � 1, and e � 0. Figure 4 shows some coexisting attractors of system (1) in some special parameter a. Figure 4(b) shows that a chaotic attractor coexists with a limit cycle. Figure 4(c) shows that two limit cycles coexist, and Figure 4(d) shows that three limit cycles (a pair of symmetric limit cycles) coexist.     (1) with One Equilibrium. For the stability of equilibrium, the Jacobian matrix of system (1) is

Chaos in System
and the characteristic polynomial is e corresponding eigenvalues for equilibrium E(0, 0, (− a/e)) are e, (a ± ������� a 2 − 4e 2 √ /2e). e types of equilibrium for different values of the parameters a and e are summarized in Table 2.
In Table 2, the types of equilibrium under different values of a and e are discussed in two cases of e > 0 and e < 0, and Tables 3 and 4 show the periodic or chaotic motions for unstable node (0 < a < 2e) and stable node-focus (0 < a < − 2e), respectively. e corresponding phase portraits and Poincaré sections are shown in Figures 5 and 6.
e Lyapunov exponents spectrum (LES) add largest Lyapunov exponent (LLE) for system (1) versus a are shown in Figure 7.

Circuit Modeling and Simulation.
In this section, the analog circuit is designed in order to illustrate the correctness of system (1). And, according to Section 2.1, fixed b � − 1, c � 1, e � 0, system (1) can be rewritten as en, analog circuit is designed by using LM741 operational amplifiers, AD633 analog multipliers, resistors, and capacitors, where the gain of multiplier AD633 is 0.1 and the power voltage of operational amplifier LM741 is E � ± 15V, and its output saturated voltage is Vsat ≈ ± 13.5V. According to system (8), the chaotic circuit is designed, as shown in Figure 8.
According to Figure 8 and Kirchhoff's law, the following circuit equation can be obtained: Owing to the limitation of LM741 and AD633 working voltages, the output voltages of the system are reduced to 1/10 of the original. Compress them according to 10: 1, and let time constant τ � t/10R 0 C 0 , then equation (9) can be expressed as  (1) with a � − 1 and b � 0; (b) chaotic attractor coexists with limit cycle in system (1) with a � − 6.17 and b � − 1; (c) two limit cycles coexist in system (1) with a � − 4.5 and b � − 1; (d) three limit cycles coexist in system (1) with a � − 0.8 and b � − 1.
From Figure 9, under the different parameters a, the rotationally symmetric chaotic attractor, symmetric limit cycle, and symmetric pair of limit cycles can be observed.
us, the circuit results are consistent with the numerical results by Figure 1 and Table 1.

Conclusion 1.
(1) When b > 0 and c > 0, at the positive or negative endpoint of the v axis, it has two semihyperbolic saddles. System (12) has many periodic solutions in the left-half plane and right-half plane (see Figure 10(a)). (2) When b < 0 and c < 0, system (12) has two saddles ( ± ��� b/c √ , 0) and six nodes (three stable nodes and three unstable nodes) at infinity in Poincaré disc. Also, it has eight separatrices which connected two saddles and six nodes at infinity (see Figure 10(b)). (3) When b < 0 and c > 0, system (12) has one stable node at the positive endpoint of the v axis and one unstable node at the negative endpoint of the v axis (see Figure 10(c)). (4) When v and c < 0, system (12) has two semihyperbolic saddles at the positive or negative endpoint of v axis and four nodes (two stable nodes and two unstable nodes) (see Figure 10(d)). e flow in the chart V 1 is the same as the flow in the chart U 1 reversing the time. Hence, the phase portrait of system (1) on the infinite sphere at the negative endpoint of the x axis is shown in Figure 10, reversing the time direction.
In the charts U 2 and V 2 , next we study dynamics of system (1) at infinity of the (HTML translation failed) axis. Taking the transformation (x, y, z) � (uw − 1 , w − 1 , vw − 1 ) and t � wτ, system (1) becomes If w � 0, system (26) reduces to Clearly, we can see that system (27) has two centers ( ± ��� c/b √ , 0) when b > 0 and c > 0 and one stable node ), and two saddles ( ± ��� c/b √ , 0) when b < 0 and c < 0. Also, it has one stable node (0, − �� − c √ ) and one unstable node (0, �� − c √ ) when b > 0 and c < 0 and no equilibrium when b < 0 and c > 0. e corresponding global phase portraits of system (27) are shown in Figure 11. e flow in the chart V 2 is the same as the flow in the local chart U 2 . Hence, the phase portrait of system (1) on the infinite sphere at the negative endpoint of the y axis is shown in Figure 11, reversing the time direction. We can summarize the global behaviors, as shown in 2.

Conclusion 2.
(1) When b > 0 and c > 0, at the positive or negative endpoint of v axis, it has two semihyperbolic saddles. System (27) has many periodic solutions in the lefthalf plane and right-half plane (see Figure 11(a)). (2) When b < 0 and c < 0, system (27) has two saddles ( ± ��� c/b √ , 0) and two nodes (0, ± �� − c √ ) (one is a stable node and the other one is an unstable node) in Poincaré disc. Also, it has two other nodes at the positive and negative endpoint of v axis. Moreover, it has eight separatrices which connect two saddles and four nodes (see Figure 11(b)). (3) When b < 0 and c > 0, system (27) has one stable node at the positive endpoint of v axis and one unstable node at the negative endpoint of v axis (see Figure 11(c)). (4) When b > 0 and c < 0, system (27) has two semihyperbolic saddles at the positive or negative endpoint of v axis and two nodes (one stable node and one unstable node) (see Figure 11(d)).
e flow in the chart V 3 is the same as the flow in the local chart U 3 . Hence, the phase portrait of system (1) on the infinite sphere at the negative endpoint of the z axis is shown in Figure 12, reversing the time direction. As in the previous analysis, we can get 3 as follows.

Periodic Solution
From Table 2, we can see that system (1) shows Hopf bifurcation at the origin when a � 0 and Hopf-zero bifurcation (also called saddle-node Hopf bifurcation or fold Hopf bifurcation) when a � e � 0. e Hopf-zero bifurcation can be considered an approximation of Hopf bifurcation when e ⟶ 0. In this section, we study the Hopf-zero bifurcation at the origin and the periodic orbit, which bifurcates in this bifurcation. Additionally, the stability of this periodic solution will be analyzed. In order to study the periodic solution of system (1), we first introduce averaging theory [48][49][50]. Lemma 1. Give the differential equation in the standard form as follows: Moreover, suppose F 1 (t, x) and F 2 (t, x, ε) are T periodic in t. Consider the averaged differential equation in D.
(2) If b > c, system (1) has a periodic solution φ(x(t, a), y(t, a), z(t, a)) with period approximately 2π of the form as follows: e solution φ is a linearly stable periodic solution.
Proof. Writing equation (1) in cylindrical coordinates (r, θ, ω) and let x � r cos θ, we get w � a + cr 2 sin 2 θ − br 2 cos 2 θ + ew. Doing the rescaling of the variables through the changes of coordinates we have Equation (37) can be written as Using the notation of the averaging theory described in Lemma 1, we have t � θ, where F 11 (θ, R, W) � RWsin 2 θ, We can immediately check that equation (38) satisfies the assumptions of Lemma 1, compute the averaged differential equation (30) with y � (R, W), and denote where Since f 11 ≡ 0, we have the real solutions (R, W) � (0, − (1/2e)) and (R, By Lemma 1, for any a > 0 sufficiently small, system (37) has a periodic solution , 0) for b > c when a tends to 0. e eigenvalues of the Jacobian matrix at the solution (0, − (1/2e)) are − (1/4e), − e. So, the periodic orbit has three stable manifolds when e > 0 or three unstable manifolds when e < 0. e eigenvalues of the Jacobian matrix at the solution (1/ , 0) are − (e/2) ± ����� e 2 + 2 √ /2. is shows that the periodic orbit has a 3D stable manifold (a generalized cylinder) and two 2D invariant manifolds (one stable and one unstable, both being cylinders) or a 3D unstable manifold (a generalized cylinder) and two 2D invariant manifolds (one stable and one unstable, both being cylinders).
For a > 0 and sufficiently small, going back to the differential system (35), we can get two periodic solutions with period approximately 2π of the form as follows: ese two periodic solutions of equation (1) become periodic solutions of period close to 2π of the form as follows: Complexity 13 x(t) � O(a), + O(a). (45) ese periodic orbits tend to the origin of coordinate when a tends to zero. erefore, they are small-amplitude periodic solutions starting at the Hopf-zero equilibrium point.
For a > 0 sufficiently small and b > c, going back to the differential system (35), we can get two periodic solutions with period approximately 2π of the form as follows: is periodic solution of equation (1) becomes periodic solutions with period close to 2π of the form as follows: is periodic orbit tends to the origin of coordinate when a tends to zero. erefore, it is a small-amplitude periodic solution starting at the Hopf-zero equilibrium point. It completes the proof of eorem 1.

Conclusion
In this paper, the studied system's chaotic dynamics with no equilibria, with an unstable node, and stable node-focus were provided. Various coexisting attractors were investigated by analytical-numerical methods, and a circuit was carried out, a good similarity between the circuit experimental results and the theoretical analysis was achieved. Furthermore, the infinity dynamics of the system were analyzed using Poincaré-Lyapunov compactification to research the global structure of system. A periodic solution which bifurcates from Hope-zero bifurcation was analyzed by the averaging theory. Indeed, the inner global dynamics and the geometrical structure of this system have been presented entirely. Furthermore, the system was time-reversible with the LEs that were symmetric around zero for some particular parameters. e relations of the time-reversible system and conservative system and the generation mechanism of chaotic dynamics are studied in future works.

Data Availability
All data, models, and code generated or used during the study are included in the submitted article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.